where \((\hat{\beta })^m_{n}\)  are the matrix element \(\left\langle m|\hat{\beta }|n\right\rangle \) 
over an orthonormal basis \(\left\langle m|n\right\rangle = \delta ^m_{n}\) , and \(z^n\) 
the complex coordinates of a normalized state
\(\left|\phi \right\rangle = \sum _n z^n \left|n\right\rangle \) , \(n\)  runs over the
dimension of the Hilbert space for the system under consideration.
The set of \(z^n\)  also serves as the homogeneous coordinates of
the projective Hilbert space as a Kähler manifold [1]}. One
can check that
\(&&f_{\!\scriptscriptstyle \hat{\beta }\hat{\gamma }} = f_{\!\scriptscriptstyle \hat{\beta }} f_{\!\scriptscriptstyle \hat{\gamma }}+ \sum _n V_{\!\scriptscriptstyle \hat{\beta }_n} V_{\!\scriptscriptstyle \hat{\gamma }_{\bar{n}}} \;,\nonumber \\&&(\hat{\beta }\hat{\gamma })^m_{n} = \sum _l (\hat{\beta })^m_{l} (\hat{\gamma })^l_{n} \;,\nonumber \\&&V_{\!\scriptscriptstyle \hat{\beta }\hat{\gamma }_n} = - f_{\!\scriptscriptstyle \hat{\beta }\hat{\gamma }} \bar{z}_n+ \sum _m \bar{z}_m (\hat{\beta }\hat{\gamma })^m_{n}\;,\) 
